# TÀI LIỆU TOÁN - WWW.MOLYMPIAD.NET - ĐỀ THI TOÁN

## $show=home 1. Which number is greater $A=\left(1+\frac{1}{2013}\right)\left(1+\frac{1}{2013^{2}}\right)\ldots\left(1+\frac{1}{2013^{n}}\right)$ where$n$is a positive integer, or${\displaystyle B=\frac{2013^{2}-1}{2012^{2}-1}}$?. 2. Given four points in the plane such that no pair of points has distance less than$\sqrt{2}$cm. Prove that there exists two of them having a distance greater than or equal to$2$cm. 3. Find the last two digits of the number $2003^{2004^{\mathstrut^{.^{.^{.^{2013}}}}}}.$ 4. Find the maximum and minimum value of the expression $P=27\sqrt{x}+8\sqrt{y}$ where$x,y$are non-negative real numbers satisfying $x\sqrt{1-y^{2}}+y\sqrt{2-x^{2}}=x^{2}+y^{2}.$ 5. Let$ABCD$be a cyclic quadrilateral, inscribed in circle$(O)$.$I$and$J$are the midpoints of$BD$and$AC$respectively. Prove that$BD$is the angle bisector of angle$AIC$if and only if$AC$is the angle bisector of angle$BJD$. 6. Solve the following system of equations $\begin{cases} x^{3}(1-x)+y^{3}(1-y) & =12xy+18\\ |3x-2y+10|+|2x-3y| & =10 \end{cases}.$ 7. Determine the greatest value of the expression $E=a^{2013}+b^{2013}+c^{2013},$ where$a,b,c$are real numbers satisfying $a+b+c=0,\quad a^{2}+b^{2}+c^{2}=1.$ 8. Let$S.ABC$be a triangular pyramid,$G$is the centroid of the base triangle$ABC$,$O$is the midpoint of$SG$. A moving plane$(\alpha)$through$O$meets the edges$SA$,$SB$,$SC$at$A',B'C'$respectively. Prove that $\frac{SA'^{2}}{AA'^{2}}+\frac{SAB'}{BB'^{2}}+\frac{SC'^{2}}{CC'^{2}}\geq\frac{AA'^{2}}{SA'^{2}}+\frac{BB'^{2}}{SB'^{2}}+\frac{CC'^{2}}{SC'^{2}}.$ 9. Find all natural numbers$n$such that $A=\left[\frac{n+3}{4}\right]+\left[\frac{n+5}{4}\right]+\left[\frac{n}{2}\right]+n^{2}+3n-1$ is a prime number, where$[x]$denotes the greatest integer not exceeding$x$. 10. Consider the real-valued function $y=a\sin(x+2013)+\cos2014x$ where$a$is given real number. Let$M,N$be the greatest and smallest values respectively of this function over$\mathbb{R}.$Prove that$M^{2}+N^{2}\geq2$. 11. Let$\{a_{n}\}$be a sequence given by $a_{1}=\frac{1}{2},\quad a_{n+1}=\frac{a_{n^{2}}}{a_{n^{2}-a_{n}+1}},\,n=1,2,\ldots$ a) Prove that the sequence$\{a_{n}\}$converges to a finite limit and find this limit. b) Let$b_{n}=a_{1}+a_{2}+\ldots+a_{n}$for each positive integer$n$. Determine the integer part$[b_{n}]$and the limit$\lim_{n\to\infty}b_{n}$. 12. Given four points$A,B,C,D$on circle$(ABC)$and$M$is a point not on this circle. Let$T_{i}$be the triangle whose three vertices are$3$of$4$given points, except point$i$($i=A,B,C,D$). Let$H_{i}$be the triangle whose vertices are the feet of the perpendicular drawn from$M$onto the edges (or extended edges) of triangles$T_{i}$($i=A,B,C,D$). Prove that a) The circumcenter of triangles$H_{i}$($i=A,B,C,D$) lie on the same circle, centered at$O'$. b) When$D$moves on the circle$(ABC)$,$O'$always lie on a fixed circle. ##$type=three$c=3$source=random$title=oot$p=1$h=1$m=hide$rm=hide ## Anniversary_$type=three$c=12$title=oot$h=1$m=hide\$rm=hide

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2006,1,2007,12,2008,12,2009,12,2010,12,2011,12,2012,12,2013,12,2014,12,2015,12,2016,12,2017,12,2018,11,2019,12,2020,12,2021,6,Anniversary,4,
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Mathematics & Youth: 2013 Issue 431
2013 Issue 431
Mathematics & Youth